Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel

Hydrodynamic focusing of cells along the region of the most rapid flow is a robust feature in downflowing suspensions of swimming gyrotactic microorganisms. Experiments performed in a downward pipe flow have reported that the focused beam-like structure of the cells is often unstable and results in the formation of regular-spaced axisymmetric blips, but the mechanism by which they are formed is not well understood. To elucidate this mechanism, in this study, we perform a linear stability analysis of a downflowing suspension of randomly swimming gyrotactic cells in a two-dimensional vertical channel. On increasing the flow rate, the basic state exhibits a focused beam-like structure. It is found that this focused beam is unstable with the varicose mode, the spatial structure, wavelength, phase speed and behaviour with the flow rate of which are remarkably similar to those of the blip instability in the pipe flow experiment. To understand the physical mechanism of the varicose mode, we perform an analysis which calculates the term-by-term contribution to the instability. It is shown that the leading physical mechanism in generating the varicose instability originates from the horizontal gradient in the cell-swimming-vector field formed by the non-uniform shear in the base flow. This mechanism is found to be supplemented by cooperation with the gyrotactic instability mechanism observed in uniform suspensions.